Mathematical analysis and numerical simulation of the motion of a glacier

Abstract

A numerical method for the simulation of the motion of a glacier in two and three dimensions is presented. Glacier ice is treated as an incompressible viscous fluid. The model equations are based on mass, momentum, energy conservation and a specific rheology law for ice (the so called Glen's flow law). In this study a simplified model due to Blatter[8], called the first order approximation (FOA), is used. The problem is split in two sub-problems: (a) the computation of the velocity field for a given glacier geometry and (b) the computation of the evolution of the glacier geometry when the velocity field is known. Concerning sub-problem (a), the FOA leads to a system of non-linear elliptic partial differential equations, which are solved numerically by a standard linear finite element method. The non-linearity is solved iteratively with a frozen coefficient method. Sub-problem (b) leads to a transport equation, which is approached with either an upwind Euler scheme or a Lax-Wendroff scheme in the two dimensional case, and with a stabilized finite element method in the three-dimensional case. All computations are done on a moving domain applying an arbitrary Lagrangian Eulerian (ALE) method. Moreover an algorithm that deforms the computational mesh in accordance with the evolution of the glacier geometry is given. From a theoretical point of view, the existence and uniqueness of a weak solution to problem (a) is established using calculus of variations methods. The solution of the discrete problem is proved to be convergent to the exact solution. Further a priori and a posteriori error estimates are given. Using several test cases, the a priori and a posteriori error estimates are numerically verified. Then simulations of the past evolution of real glaciers is done and the computations are compared with measurements as well as with results of simulations found in the literature. Finally, under the assumption that the climate stays the same over the next 20 years, some predictions of the future evolution of these glaciers are presented.